Subgroups — #33057454

Order 18688.

652 conjugacy classes of subgroups.

The subgroup lattice has 340 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22618949	C₂	1	2	yes	maximal, complemented
2	C₄×#14556613	C₂	1	1	yes	maximal, characteristic
3	#22608817	C₂	1	2	yes	maximal, complemented
4	#22608776	C₂	1	4	yes	maximal, complemented
5	C₈×Dic₂₉₂	C₂	1	2	yes	maximal, complemented
6	#22608694	C₂	1	1	yes	maximal, characteristic, complemented
7	#22608366	C₂	1	2	yes	maximal, complemented
8	C₈×M₁₆×C₇₃	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#14556613	C₄	1	1	yes	characteristic
10	C₄×#11924028	C₂²	1	2	yes	—
11	C₈×Dic₁₄₆	C₂²	1	4	yes	—
12	C₄×Dic₂₉₂	C₄	1	1	yes	characteristic
13	C₄×Dic₂₉₂	C₂²	1	1	yes	characteristic
14	C₄×#11924017	C₄	1	2	yes	—
15	C₄×#11924017	C₂²	1	2	yes	—
16	C₄×D₅₈₄	C₄	1	1	yes	characteristic
17	C₄×D₅₈₄	C₂²	1	1	yes	characteristic
18	#17988666	C₂²	1	4	yes	—
19	#17988663	—	2	4	no	—
20	#17988644	C₂²	1	2	yes	—
21	#17988642	—	2	2	no	—
22	#17988625	C₄	1	1	yes	characteristic
23	#17988584	C₄	1	2	yes	—
24	#17988539	—	2	2	no	—
25	#17988340	C₂²	1	2	yes	—
26	#17988338	C₂²	1	2	yes	—
27	#17988319	C₂²	1	4	yes	—
28	#17988316	C₂²	1	4	yes	—
29	#431×C₇₃	C₂²	1	1	yes	characteristic
30	C₄×M₁₆×C₇₃	C₂²	1	1	yes	characteristic
31	C₂×C₄×C₈×C₇₃	C₂²	1	1	yes	characteristic
32	#322×C₇₃	C₂²	1	2	yes	—
33	C₈²×C₇₃	C₂²	1	2	yes	—
34	C₂×#11924028	C₂×C₄	1	2	yes	—
35	#14556613	—	2	4	no	—
36	C₂×Dic₂₉₂	C₂×C₄	1	1	yes	characteristic
37	C₂×#11924017	C₂×C₄	1	2	yes	—
38	C₂×D₅₈₄	C₂×C₄	1	1	yes	characteristic
39	C₂×#11924011	—	2	4	no	—
40	C₄×Dic₁₄₆	C₂×C₄	1	2	yes	—
41	C₄×Dic₁₄₆	C₂³	1	2	yes	—
42	C₄×D₂₉₂	C₂×C₄	1	2	yes	—
43	C₄×D₂₉₂	C₂³	1	2	yes	—
44	C₄×C₂²⋊D₇₃	—	2	2	no	—
45	C₄×C₂²⋊D₇₃	—	2	2	no	—
46	#14556550	C₂×C₄	1	2	yes	—
47	#14556548	—	2	2	no	—
48	C₄²×D₇₃	—	2	2	no	—
49	#14556525	C₂×C₄	1	1	yes	characteristic
50	#14556525	C₂×C₄	1	1	yes	characteristic
51	#14556523	C₂×C₄	1	2	yes	—
52	C₄⋊#10596831	C₂³	1	4	yes	—
53	#14556520	—	2	4	no	—
54	#14556516	C₂×C₄	1	2	yes	—
55	#14556516	C₂×C₄	1	2	yes	—
56	#14556513	C₂×C₄	1	2	yes	—
57	#14556513	C₂×C₄	1	2	yes	—
58	C₂²⋊#10596831	—	2	2	no	—
59	#14556503	—	2	4	no	—
60	#14556502	—	2	2	no	—
61	C₈×Dic₇₃	—	2	4	no	—
62	C₂×M₁₆×C₇₃	D₄	1	2	yes	—
63	C₂²×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
64	C₂²×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
65	C₂×C₄²×C₇₃	C₂³	1	1	yes	characteristic
66	C₄⋊C₈×C₇₃	—	2	1	no	—
67	C₂²⋊C₈×C₇₃	—	2	1	no	—
68	#97×C₇₃	D₄	1	1	yes	characteristic
69	#97×C₇₃	D₄	1	1	yes	characteristic
70	C₄×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
71	C₄×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
72	C₄×C₈×C₇₃	C₂³	1	2	yes	—
73	C₄×C₈×C₇₃	C₂³	1	1	yes	characteristic
74	C₄×C₈×C₇₃	C₂³	1	1	yes	characteristic
75	C₄×C₈×C₇₃	C₂³	1	2	yes	—
76	#11924028	—	2	4	no	—
77	#11924028	—	4	2	no	—
78	C₂×Dic₁₄₆	C₂²×C₄	1	2	yes	—
79	C₂×D₂₉₂	C₂²×C₄	1	2	yes	—
80	C₂×C₂²⋊D₇₃	—	2	2	no	—
81	C₂×C₄×D₇₃	—	2	2	no	—
82	C₂×C₄×D₇₃	—	2	2	no	—
83	C₂×C₄×D₇₃	—	2	2	no	—
84	Dic₂₉₂	M₁₆	1	4	yes	complemented
85	#11924017	—	2	4	no	—
86	D₅₈₄	—	2	2	no	—
87	#11924011	—	4	8	no	—
88	C₂×#10596831	—	2	4	no	—
89	C₄⋊Dic₇₃	C₂²×C₄	1	2	yes	—
90	C₄⋊Dic₇₃	C₂²×C₄	1	2	yes	—
91	#11924007	—	2	2	no	—
92	#11924007	—	2	2	no	—
93	C₂²⋊Dic₇₃	—	2	2	no	—
94	#11924005	—	2	2	no	—
95	#11924005	—	2	2	no	—
96	C₄×Dic₇₃	—	2	2	no	—
97	C₄×Dic₇₃	—	2	2	no	—
98	C₄×Dic₇₃	—	2	2	no	—
99	C₂²×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
100	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
101	C₂²×C₄×C₇₃	#41	1	1	yes	characteristic
102	M₁₆×C₇₃	—	2	4	no	—
103	C₂×C₈×C₇₃	—	2	2	no	—
104	C₂×C₈×C₇₃	—	2	2	no	—
105	C₂×C₈×C₇₃	—	4	1	no	—
106	C₂×C₈×C₇₃	C₂²×C₄	1	1	yes	characteristic
107	C₂×C₈×C₇₃	C₂²×C₄	1	1	yes	characteristic
108	C₂×C₈×C₇₃	C₂²×C₄	1	1	yes	characteristic
109	C₂×C₈×C₇₃	C₂²×C₄	1	1	yes	characteristic
110	C₂×C₈×C₇₃	C₂×D₄	1	2	yes	—
111	C₂×C₈×C₇₃	C₂×D₄	1	2	yes	—
112	C₄²×C₇₃	C₂²×C₄	1	1	yes	characteristic
113	C₄²×C₇₃	C₂×D₄	1	1	yes	characteristic
114	C₄²×C₇₃	#41	1	1	yes	characteristic
115	C₄²×C₇₃	C₂⁴	1	1	yes	characteristic
116	C₂³×C₇₃	C₄×D₄	1	1	yes	characteristic
117	C₂²×D₇₃	—	2	2	no	—
118	C₂×C₄×C₇₃	—	2	1	no	—
119	C₂×C₄×C₇₃	—	2	1	no	—
120	C₂×C₄×C₇₃	—	2	2	no	—
121	C₂×C₄×C₇₃	—	4	1	no	—
122	C₂×C₄×C₇₃	C₄×D₄	1	1	yes	characteristic
123	C₂×C₄×C₇₃	C₄×D₄	1	1	yes	characteristic
124	C₂×C₄×C₇₃	C₄×D₄	1	1	yes	characteristic
125	C₂×C₄×C₇₃	C₂³×C₄	1	1	yes	characteristic
126	C₂×C₄×C₇₃	C₂²×D₄	1	1	yes	characteristic
127	C₂×C₄×C₇₃	C₂×#41	1	1	yes	characteristic
128	C₂²⋊D₇₃	—	4	4	no	—
129	C₂×Dic₇₃	—	2	2	no	—
130	C₂×Dic₇₃	—	2	2	no	—
131	C₂×Dic₇₃	—	2	2	no	—
132	D₂₉₂	—	2	2	no	—
133	D₂₉₂	—	2	2	no	—
134	C₄×D₇₃	—	4	4	no	—
135	C₄×D₇₃	—	4	4	no	—
136	C₄×D₇₃	—	8	2	no	—
137	Dic₁₄₆	—	2	2	no	—
138	Dic₁₄₆	C₂×M₁₆	1	4	yes	—
139	C₈×C₇₃	—	2	1	no	—
140	C₈×C₇₃	—	2	1	no	—
141	C₈×C₇₃	—	2	1	no	—
142	C₈×C₇₃	—	2	2	no	—
143	C₈×C₇₃	—	4	2	no	—
144	C₈×C₇₃	C₂×M₁₆	1	2	yes	—
145	C₈×C₇₃	#135	1	4	yes	complemented
146	#10596831	—	4	4	no	—
147	C₂²×C₇₃	—	2	1	no	—
148	C₂²×C₇₃	—	2	1	no	—
149	C₂²×C₇₃	—	2	1	no	—
150	C₂²×C₇₃	C₂×C₄×D₄	1	1	yes	characteristic
151	C₂×D₇₃	—	4	2	no	—
152	C₂×D₇₃	—	4	2	no	—
153	C₂×D₇₃	—	4	2	no	—
154	C₄×C₇₃	—	2	1	no	—
155	C₄×C₇₃	—	2	1	no	—
156	C₄×C₇₃	—	2	1	no	—
157	C₄×C₇₃	—	2	1	no	—
158	C₄×C₇₃	C₂²×M₁₆	1	1	yes	characteristic
159	C₄×C₇₃	#568	1	1	yes	characteristic
160	C₄×C₇₃	C₂×#135	1	2	yes	—
161	Dic₇₃	—	2	4	no	—
162	Dic₇₃	—	4	2	no	—
163	Dic₇₃	—	4	2	no	—
164	#18039	—	73	1	no	maximal, Sylow
165	C₂×C₇₃	—	4	1	no	—
166	C₂×C₇₃	D₄×M₁₆	1	1	yes	characteristic
167	C₂×C₇₃	C₄×#135	1	1	yes	characteristic
168	C₂×C₇₃	#2957	1	1	yes	characteristic
169	D₇₃	—	8	2	no	—
170	#2971	—	73	2	no	—
171	C₄×#135	—	73	1	no	—
172	#1589	—	73	2	no	—
173	#1583	—	73	4	no	—
174	C₈×Q₁₆	—	73	2	no	—
175	#1568	—	73	1	no	—
176	#1488	—	73	2	no	—
177	C₈×M₁₆	D₇₃	1	1	yes	characteristic, complemented
178	C₇₃	#18039	1	1	yes	characteristic, complemented, Sylow
179	C₂×#135	—	73	1	no	—
180	C₄×#41	—	73	2	no	—
181	C₈×Q₈	—	73	4	no	—
182	C₄×Q₁₆	—	73	1	no	—
183	C₄×Q₁₆	—	73	1	no	—
184	C₄×SD₁₆	—	73	2	no	—
185	C₄×SD₁₆	—	73	2	no	—
186	C₄×D₈	—	73	1	no	—
187	C₄×D₈	—	73	1	no	—
188	#436	—	73	4	no	—
189	#436	—	146	4	no	—
190	#432	—	73	2	no	—
191	#432	—	146	2	no	—
192	#431	C₂×D₇₃	1	1	yes	characteristic
193	#427	—	73	1	no	—
194	#416	—	73	2	no	—
195	C₄×M₁₆	—	146	2	no	—
196	C₄×M₁₆	C₂×D₇₃	1	1	yes	characteristic
197	C₂×C₄×C₈	C₂×D₇₃	1	1	yes	characteristic
198	#335	—	73	2	no	—
199	#334	—	73	2	no	—
200	Q₈⋊C₈	—	73	4	no	—
201	D₄⋊C₈	—	73	4	no	—
202	#322	C₂×D₇₃	1	2	yes	—
203	C₈²	C₂×D₇₃	1	2	yes	—
204	C₂×#41	—	73	2	no	—
205	#135	—	146	4	no	—
206	C₂×Q₁₆	—	73	1	no	—
207	C₂×SD₁₆	—	73	2	no	—
208	C₂×D₈	—	73	1	no	—
209	C₂×M₁₆	—	146	4	no	—
210	C₂×M₁₆	D₂₉₂	1	2	yes	—
211	C₂²×C₈	C₄×D₇₃	1	1	yes	characteristic
212	C₂²×C₈	C₄×D₇₃	1	1	yes	characteristic
213	C₄×Q₈	—	73	2	no	—
214	C₄×Q₈	—	73	2	no	—
215	C₄×D₄	—	73	2	no	—
216	C₄×D₄	—	73	2	no	—
217	C₄×D₄	—	146	2	no	—
218	C₄×D₄	—	146	2	no	—
219	#117	—	73	2	no	—
220	#117	—	146	2	no	—
221	C₂×C₄²	—	146	2	no	—
222	C₂×C₄²	C₂²×D₇₃	1	1	yes	characteristic
223	#107	—	73	1	no	—
224	#107	—	73	1	no	—
225	#106	—	73	2	no	—
226	C₄⋊C₈	—	2	1	no	—
227	C₄⋊C₈	—	73	4	no	—
228	C₄⋊C₈	—	146	4	no	—
229	#103	—	73	2	no	—
230	#103	—	73	2	no	—
231	#102	—	73	2	no	—
232	#102	—	73	2	no	—
233	C₂²⋊C₈	—	2	1	no	—
234	C₂²⋊C₈	—	146	2	no	—
235	C₂²⋊C₈	—	146	4	no	—
236	#97	—	146	2	no	—
237	#97	D₂₉₂	1	1	yes	characteristic
238	#97	D₂₉₂	1	1	yes	characteristic
239	C₄×C₈	—	146	4	no	—
240	C₄×C₈	C₄×D₇₃	1	1	yes	characteristic
241	C₄×C₈	C₄×D₇₃	1	1	yes	characteristic
242	C₄×C₈	C₂²×D₇₃	1	2	yes	—
243	C₄×C₈	C₂²×D₇₃	1	1	yes	characteristic
244	C₄×C₈	C₂²×D₇₃	1	1	yes	characteristic
245	C₄×C₈	C₂²×D₇₃	1	2	yes	—
246	#41	—	292	2	no	—
247	#41	—	146	4	no	—
248	C₂×Q₈	—	73	2	no	—
249	C₂×D₄	—	73	2	no	—
250	C₂×D₄	—	146	2	no	—
251	C₂²×C₄	—	146	2	no	—
252	C₂²×C₄	—	146	2	no	—
253	C₂²×C₄	—	146	2	no	—
254	C₂²×C₄	C₂×C₄×D₇₃	1	1	yes	characteristic
255	C₂²×C₄	C₂×D₂₉₂	1	1	yes	characteristic
256	C₂²×C₄	#11924028	1	1	yes	characteristic
257	Q₁₆	—	73	4	no	—
258	SD₁₆	—	146	4	no	—
259	D₈	—	146	2	no	—
260	M₁₆	—	2	4	no	—
261	M₁₆	—	292	8	no	—
262	C₂×C₈	—	2	2	no	—
263	C₂×C₈	—	2	2	no	—
264	C₂×C₈	—	4	1	no	—
265	C₂×C₈	—	146	4	no	—
266	C₂×C₈	C₂×C₄×D₇₃	1	1	yes	characteristic
267	C₂×C₈	C₂×C₄×D₇₃	1	1	yes	characteristic
268	C₂×C₈	C₂×C₄×D₇₃	1	1	yes	characteristic
269	C₂×C₈	C₂×C₄×D₇₃	1	1	yes	characteristic
270	C₂×C₈	C₂×D₂₉₂	1	2	yes	—
271	C₂×C₈	C₂×D₂₉₂	1	2	yes	—
272	C₄⋊C₄	—	73	2	no	—
273	C₄⋊C₄	—	73	2	no	—
274	C₄⋊C₄	—	146	2	no	—
275	C₄⋊C₄	—	146	2	no	—
276	C₂²⋊C₄	—	146	2	no	—
277	C₂²⋊C₄	—	146	2	no	—
278	C₂²⋊C₄	—	146	2	no	—
279	C₄²	—	146	2	no	—
280	C₄²	—	146	2	no	—
281	C₄²	—	146	2	no	—
282	C₄²	C₂×C₄×D₇₃	1	1	yes	characteristic
283	C₄²	C₂×D₂₉₂	1	1	yes	characteristic
284	C₄²	#11924028	1	1	yes	characteristic
285	C₄²	C₂³×D₇₃	1	1	yes	characteristic
286	C₂³	—	146	2	no	—
287	C₂³	C₄×D₂₉₂	1	1	yes	characteristic
288	Q₈	—	73	4	no	—
289	Q₈	—	146	2	no	—
290	D₄	—	292	4	no	—
291	D₄	—	146	2	no	—
292	D₄	—	146	2	no	—
293	C₂×C₄	—	2	1	no	—
294	C₂×C₄	—	2	1	no	—
295	C₂×C₄	—	2	2	no	—
296	C₂×C₄	—	4	1	no	—
297	C₂×C₄	—	292	4	no	—
298	C₂×C₄	—	292	4	no	—
299	C₂×C₄	—	584	2	no	—
300	C₂×C₄	—	146	2	no	—
301	C₂×C₄	—	146	2	no	—
302	C₂×C₄	—	146	2	no	—
303	C₂×C₄	C₄×D₂₉₂	1	1	yes	characteristic
304	C₂×C₄	C₄×D₂₉₂	1	1	yes	characteristic
305	C₂×C₄	C₄×D₂₉₂	1	1	yes	characteristic
306	C₂×C₄	C₂²×C₄×D₇₃	1	1	yes	characteristic
307	C₂×C₄	C₂²×D₂₉₂	1	1	yes	characteristic
308	C₂×C₄	C₂×#11924028	1	1	yes	characteristic
309	C₈	—	2	1	no	—
310	C₈	—	2	1	no	—
311	C₈	—	2	1	no	—
312	C₈	—	2	2	no	—
313	C₈	—	4	2	no	—
314	C₈	—	292	4	no	—
315	C₈	M₁₆×D₇₃	1	2	yes	—
316	C₈	#14556613	1	4	yes	complemented
317	C₂²	—	2	1	no	—
318	C₂²	—	2	1	no	—
319	C₂²	—	2	1	no	—
320	C₂²	—	292	2	no	—
321	C₂²	—	292	2	no	—
322	C₂²	—	292	2	no	—
323	C₂²	C₂×C₄×D₂₉₂	1	1	yes	characteristic
324	C₄	—	2	1	no	—
325	C₄	—	2	1	no	—
326	C₄	—	2	1	no	—
327	C₄	—	2	1	no	—
328	C₄	—	292	2	no	—
329	C₄	—	292	2	no	—
330	C₄	—	146	4	no	—
331	C₄	C₂×M₁₆×D₇₃	1	1	yes	characteristic
332	C₄	#17989353	1	1	yes	characteristic
333	C₄	C₂×#14556613	1	2	yes	—
334	C₂	—	4	1	no	—
335	C₂	—	584	2	no	—
336	C₂	M₁₆×D₂₉₂	1	1	yes	characteristic
337	C₂	C₄×#14556613	1	1	yes	characteristic
338	C₂	#22618849	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²×D₂₉₂
commutator	C₂×C₄×C₇₃	C₂³×C₄
Frattini	C₄²	C₂³×D₇₃
Fitting	C₈×M₁₆×C₇₃	C₂
socle	C₂²×C₇₃	C₂×C₄×D₄
prime radical	C₂×#14556613	C₄
cosocle	C₄²×C₇₃	C₂⁴
hypercenter	C₈×M₁₆	D₇₃
hypocenter	C₇₃	#18039
LCS	G ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂×C₄ ◁ C₄×M₁₆ ◁ C₈×M₁₆

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
1. #22618949	C₂	semidirect
3. #22608817	C₂	semidirect
4. #22608776	C₂	semidirect
5. C₈×Dic₂₉₂	C₂	semidirect
6. #22608694	C₂	semidirect
7. #22608366	C₂	semidirect
8. C₈×M₁₆×C₇₃	C₂	semidirect
84. Dic₂₉₂	M₁₆	semidirect
145. C₈×C₇₃	#135	semidirect
177. C₈×M₁₆	D₇₃	semidirect
178. C₇₃	#18039	semidirect
316. C₈	#14556613	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
#22618949	2
C₄×#14556613	2
#22608817	2
#22608776	2
C₈×Dic₂₉₂	2
#22608694	2
#22608366	2
C₈×M₁₆×C₇₃	2
#18039	73